where a and r are the only two variables. What exactly does the #1 mean here? To which variables does it refer? How do I make Mathematica replace all these place holders with its corresponding variable?

Use e.g. Root[a^3 #1^2 + a^2 #1^4 - 22 a #1^6 + #1^8 &, 5] // ToRadicals. However in general one cannot express Root objects in terms of radicals since it is mathematically impossible. See e.g. How do I work with Root objects?
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ArtesJul 12 '13 at 11:20

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